January 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Real index Bernoulli numbers?
by Steve Witham 09 Jan '15

09 Jan '15

Can't these be used to sum real, complex and negative powers (except -1) of sequences of integers, as with Faulhaber's formula? For "sequence lengths" that are real, negative or complex, too? All the variations of Faulhaber's formula seem to break down when the power is -1, but in just that case there's a generalization of the harmonic numbers H(n) for complex n. --Steve From: Daniel Asimov <asimov(a)msri.org> Thanks, Mike. That's a very nice paper! --Dan > On Jan 6, 2015, at 12:04 PM, Mike Stay<metaweta(a)gmail.com> wrote: > > http://arxiv.org/abs/physics/9705021 > > On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov<dasimov(a)earthlink.net> wrote: >> Are there extensions of Bernoulli numbers to real (or complex) index?

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[math-fun] RE Chess Problem (H Baker)
by Guy Haworth 09 Jan '15

09 Jan '15

The two-phase chess 'game' seems to less than well defined, and - as far as it is defined - seems to be a puzzle rather than a game as there is only one results ... 'draw'. Losing Chess, which prioritises losing and capturing men, is perhaps the nearest in concept. There are various variants of it, but recent work on the mainline version of Losing Chess (ICGA Journal, Vol. 37-2 (2014) ) indicates that 1.e3 is a win for White. Guy

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[math-fun] Chess problem
by Henry Baker 09 Jan '15

09 Jan '15

Does the following form of chess have a name? The game consists of 2 phases: Phase 1. Both players conspire to move their pieces in such a way to get ready for phase 2, but there are _no_ captures allowed in phase 1. (As usual, the king cannot move into check, and must immediately move out of check.) Phase 2. The only moves allowed are _captures_, so each move has to set up (or at least not remove the possibility) for the other side to also capture. The game ends with both sides having only their king left, hence it is a draw. I got the idea for this game after listening to a description of WWI, so I would call it WWI "massacre" chess.

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[math-fun] 1000 billion digits of Pi in 1 page
by Simon Plouffe 08 Jan '15

08 Jan '15

Hello, I finally managed to organize the page on the server, http://plouffe.fr/piat1000billion/index.html there are 10000 images of 10000x10000 in one single page. one trick : you can use the mouse to zoom out the whole page with the 'CTRL' key and the 'roulette'. any patterns ? : not so far. this combined image has 1000 billion pixels. Best regards, Simon Plouffe

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Re: [math-fun] Real index Bernoulli numbers?
by Bill Gosper 08 Jan '15

08 Jan '15

On 2015-01-06 12:58, Daniel Asimov wrote: > Thanks, Mike. That's a very nice paper! > > --Dan > > >> On Jan 6, 2015, at 12:04 PM, Mike Stay <metaweta(a)gmail.com> wrote: >> >> http://arxiv.org/abs/physics/9705021 >> >> On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov <dasimov(a)earthlink.net> wrote: >>> Are there extensions of Bernoulli numbers to real (or complex) index? >>> >>> Is there one that is generally accepted as the natural way to do it? >>> >>> It would be nice if there were a natural real-index Bernoulli function that took real values. >>> >>> I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used? >>> >>> --Dan <Grumble>: a. People will never agree whether B_n is B_n(0) or B_n(1), i.e. B_1 = -1/2 or 1/2. b. In a formula, it is nearly always possible to generalize B_n to B_n(a), the Bernoulli polynomial. So change the question to "What is B_s(a)?". Obvious definition: BernoulliB[s, a]:= -s HurwitzZeta[1 - s, a] . This is numerically consistent with my occasionally repeated 3×3 product: MatrixForm[{{0, 0, BernoulliB[n, x]/m}, {0, 0, HarmonicNumber[m]/m}, {0, 0, 1}}] == Product[MatrixForm[{{(k - m)/(k + 1), (k - m)*(x + k - 1)^n/(k + 1), (x + k - 1)^n/k}, {0, (k - m)/(k + 1), 1/k}, {0, 0,1}}], {k, m}] BernoulliB[n, x] 0 0 ---------------- m HarmonicNumber[m] [0 0 -----------------] == m 0 0 1 n n k - m (k - m) (-1 + k + x) (-1 + k + x) ----- --------------------- ------------- 1 + k 1 + k k k - m 1 Product[ 0 ----- - , {k, m}] 1 + k k 0 0 1 at least for integer n and any m > n. Empirically, for noninteger n, this converges to the HurwitzZeta definition as m increases. To test this, say for m=3 In[103]:= %84 /. Product[a_, b_] :> Dot @@ (Table[a[[1]], b] /. m -> 3) /. m -> 3 During evaluation of In[103]:= Table::iterb: Iterator {k,m} does not have appropriate bounds. >> Out[103]= (0 0 1/3 BernoulliB[n,x] 0 0 11/18 0 0 1 )=={{0,0,x^n/2-1/2 (1+x)^n+1/9 (2+x)^n+1/3 (x^n/3+1/3 (1+x)^n)},{0,0,11/18},{0,0,1}} In[104]:= FullSimplify[Table[% /. MatrixForm -> Identity, {n, 0, 3}]] Out[104]= {True, True, True, False} (For {True, True, True,True}, we'd need m>3.) Using BernoulliB[n,1] = -n Zeta[1-n], m = 22 gives -n Zeta[1 - n] -> (1/232792560)(859193865 - 4262242455 2^n - 299075781161 2^n 3^(1 + n) + 2513922025 3^(2 + n) + 1877016414259 3^(1 + 2 n) - 96874882575 4^n - 613864407 4^(1 + n) 5^n + 13153780461 5^(2 + n) + 1997664536397 7^n + 77765480 3^(1 + n) 7^n - 496222352041 2^n 7^(1 + n) - 3673960371123 8^n - 19547829457 2^(2 + n) 9^n - 7235136890023 10^n + 7818300885353 11^n - 1322685 2^(3 + n) 11^n - 7110728313367 12^n + 5433803166113 13^n + 1843591886513 15^n - 803166403663 16^n + 282418051292 17^n + 16412991972 19^n) Plotting Subtract@@% {n,0,7} stays between -.01 and .003, and is of course 0 at integer n. --rwg

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Re: [math-fun] math-fun Digest, Vol 143, Issue 10
by Steve Witham 08 Jan '15

08 Jan '15

On 1/6/15 2:00 PM, math-fun-request(a)mailman.xmission.com wrote: > From: Whitfield Diffie <whitfield.diffie(a)gmail.com> > Subject: Re: [math-fun] NAND flash memory > >> ... an erase operation on such a block resets the block to all 1's. > If erase resents the bits of the block to all the same value, why > not call it zero? > I guess it's like why they call it "NAND" flash instead of NOR flash (there is NOR flash already): they're referring to some particular point in the circuit being at the conventional voltage for "1". (And the conventional meaning of a NAND gate vs. NOR gate, even though the two do the same job, just inverted.) Similarly, lines within or between chips sometimes have names like "not-reset", because the engineers have to know which voltage (the low one) means reset. In old TTL chips, making something zero took power; if you left a line disconnected, a resistor on the input would pull it up to one. Probably at some level of the software of the control computer on the chip, the meaning gets inverted, so that as a user you can say, "clear all blocks to zero" and the corresponding physical blocks get "reset to 1." Again, just guessing. That weird overlap between electrical and actual boolean thinking always used to bother me. --Steve

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Re: [math-fun] Real index Bernoulli numbers?
by Warren D Smith 08 Jan '15

08 Jan '15

> Are there extensions of Bernoulli numbers to real (or complex) index? > > Is there one that is generally accepted as the natural way to do it? > > It would be nice if there were a natural real-index Bernoulli function that > took real values. > > I read somewhere that Ramanujan used to refer to real order Bernoulli > numbers in his notebooks. Does anyone know what definition he used? --The generating function for Bernoulli(n)/n! is z/[exp(z)-1]. So we can use Cauchy residue theorem to write Bernoulli(n) = Gamma(n+1)/(2*pi*i) (* int z^(1-n) / [exp(z)-1] dz where the contour of integration encircles the origin going counterclockwise and is contained within the strip |Im(z)|<2*pi. Actually, it need not be contained in this strip, if it is shaped right. All we really need is that |Im()|<2*pi when Re(z)=0 and that if the contour goes to infinity it must do so in good directions. I would suggest a parabola-like path which starts at someplace around +i+RealInfinity, stays above the positive real axis, crosses the negative real axis, then staying below the positive real axis goes to someplace near -i+RealInfinity. That way we do not care if the integrand is only defined on a complex z-plane SLIT along the positive real axis. Define the ln function in this slit way and the power function via A^B = exp(B*ln(A)). And then observe that this definition will produce a unique Bernoulli(n) value when n is any complex number, not required to be an integer. Furthermore by reflection symmetry of everything about the real axis we see it always produces real values if n is real because the imaginary part cancels out. --Warren D. Smith.

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[math-fun] movies Re: Real index Bernoulli numbers?
by Steve Witham 08 Jan '15

08 Jan '15

The movie of "scattering" of the zeros is arCHived: https://web.archive.org/web/19991013213431/http://www.damtp.cam.ac.uk/user/… The "codes" seem to be here: https://web.archive.org/web/19991105024407/http://www.damtp.cam.ac.uk/user/… Other stuff, including more movies, that used to be in that directory: https://web.archive.org/web/*/http://www.damtp.cam.ac.uk/user/scw21/papers/… For each of those, you must then pick a date. Generally the latest fetches got "Permission denied" errors, so I picked 1999 copies above. --Steve From: Daniel Asimov <asimov(a)msri.org> Thanks, Mike. That's a very nice paper! --Dan > On Jan 6, 2015, at 12:04 PM, Mike Stay<metaweta(a)gmail.com> wrote: > > http://arxiv.org/abs/physics/9705021 > > On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov<dasimov(a)earthlink.net> wrote: >> Are there extensions of Bernoulli numbers to real (or complex) index?

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[math-fun] Humongous Thud coming
by Warren D Smith 07 Jan '15

07 Jan '15

http://www.nytimes.com/2015/01/08/science/in-a-far-off-galaxy-2-black-holes… https://www.caltech.edu/news/unusual-light-signal-yields-clues-about-elusiv… Matthew J. Graham, S. G. Djorgovski, Daniel Stern, Eilat Glikman, Andrew J. Drake, Ashish A. Mahabal, Ciro Donalek, Steve Larson, Eric Christensen: A possible close supermassive black-hole binary in a quasar with optical periodicity, NATURE Wednesday 7 January 2015 http://www.nature.com/nature/journal/vaop/ncurrent/full/nature14143.html If they are right, then two supermassive black holes are in an orbit the size of the solar system, and are going to merge soon (within 1 million years) due to loss of energy from gravity waves it is radiating. The resulting thud is going to be far bigger than anything humanity has ever observed. Matthew J. Graham, S. G. Djorgovski, Daniel Stern, Eilat Glikman, Andrew J. Drake, Ashish A. Mahabal, Ciro Donalek, Steve Larson, Eric Christensen, DOI:10.1038/nature14143

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[math-fun] denesting mumbles
by Bill Gosper 06 Jan '15

06 Jan '15

√(1 - ⁿ√R) denests when a certain polynomial P_n(R,x) factors. E.g. P₄(R,x) =1 - 256 x^2 + 128 R x^2 + 1024 R x^4 + 3200 R^2 x^4 - 8192 R^3 x^6 - 28672 R^4 x^8 + 34816 R^5 x^8 - 131072 R^6 x^10 + 131072 R^7 x^12 + 1048576 R^10 x^16 . These grow *fast* with n. I'm really interested in n=6 and 7, for each of which I have one "splitting" R. So I decided to collect a bunch of "toy" R for n=4 from old Macsyma notebooks. I can't find them!! I did find a small one converted to an html. It gave R= 64512/260144641 In[748]:= Factor[%728[[1]] /. R -> %] (using a[3] for x) Out[748]= ((-33038369407 + 458752 a[3]^2) (-33038369407 + 589824 a[3]^2) (1091533853073393531649 + 33569097773154048 a[3]^2 + 262193283072 a[ 3]^4) (1191446152405248657777607437681912764659201 - 304972394845902975940055612302553191173586944 a[3]^2 + 302486329754988497319546160627366515179520 a[3]^4 + 18453680701462988948598008315904 a[ 3]^8))/14195439340812710119888169814159419160980592549659303797\ 19569954760828848317673958401 (How the he<< did I find that?) Ok, let's just try a million R and look for a pattern: Do[If[Mod[i, 10000] == 0, Print[i]]; If[{} =!= #, Print[pos2newman[i], #]] &@ Cases[Factor[%728[[1]] /. R -> pos2newman[i]], p_Plus /; Length[p] == 2], {i, 1, 999999}] // tim 10000 20000 ... 890000 900000 Nada! OK, how long do we need to wait for Mr. Newman to reach 64512/260144641? In[742]:= newman2pos[64512/260144641]//Short Out[742]//Short= 3490308671510526617822104163<<6014>>9495630609150025673335635968 In[743]:= N[%] Out[743]= 3.490308671510527*10^6069 A mere googol^60. Maybe I'm just impatient, but I'd really like a newmanoid bijection with more highly composite rationals. E.g. 69 ↔︎ {1,0,-2,1}, where the list is exponents of 2,3,5,7,... . Or better yet, somebody just tell me how to find the magic Rs. --Bill

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